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Hilbert symbol

Hilbert symbol

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In mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, given a local field
Local field
In mathematics, a local field is a special type of field that is a locally compact topological field with respect to a non-discrete topology.Given such a field, an absolute value can be defined on it. There are two basic types of local field: those in which the absolute value is archimedean and...

 K, such as the fields of reals
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

 or p-adic number
P-adic number
In mathematics, and chiefly number theory, the p-adic number system for any prime number p extends the ordinary arithmetic of the rational numbers in a way different from the extension of the rational number system to the real and complex number systems...

s, whose multiplicative group
Multiplicative group
In mathematics and group theory the term multiplicative group refers to one of the following concepts, depending on the context*any group \scriptstyle\mathfrak \,\! whose binary operation is written in multiplicative notation ,*the underlying group under multiplication of the invertible elements of...

 of non-zero elements is K×, the Hilbert symbol is an algebraic construction, extracted from reciprocity laws, and important in the formulation of local class field theory. As the name suggests, it was in some sense introduced by David Hilbert
David Hilbert
David Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...

, although it would be anachronistic to say that of the local field formulation.

Explicitly, it is the function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

 (–, –) from K× × K× to {−1,1} defined by

Properties


The following three properties follow directly from the definition, by choosing suitable solutions of the diophantine equation above:
  • If a is a square, then (a, b) = 1 for all b.
  • For all a,b in K×, (a, b) = (b, a).
  • For any a in K× such that a−1 is also in K×, we have (a, 1−a) = 1.

The (bi)multiplicativity, i.e., = (a, b1)·(a, b2)
for any a, b1 and b2 in K× is, however, more difficult to prove, and requires the development of local class field theory.

The third property ensures that the Hilbert symbol factors over the second Milnor K-group
Milnor K-theory
In mathematics, Milnor K-theory was an early attempt to define higher algebraic K-theory, introduced by .The calculation of K2 of a field k led Milnor to the following ad hoc definition of "higher" K-groups by...

 , which is by definition
K×K× / (a ⊗ 1−a, aK× \ {1})

By the first property it even factors over . This is the first step towards the Milnor conjecture
Milnor conjecture
In mathematics, the Milnor conjecture was a proposal by of a description of the Milnor K-theory of a general field F with characteristic different from 2, by means of the Galois cohomology of F with coefficients in Z/2Z. It was proved by .-Statement of the theorem:Let F be a field of...

.

Interpretation as an algebra


The Hilbert symbol can also be used to denote the central simple algebra
Central simple algebra
In ring theory and related areas of mathematics a central simple algebra over a field K is a finite-dimensional associative algebra A, which is simple, and for which the center is exactly K...

 over K with basis 1,i,j,k and multiplication rules , , . In this case the algebra represents an element of order 2 in the Brauer group
Brauer group
In mathematics, the Brauer group of a field K is an abelian group whose elements are Morita equivalence classes of central simple algebras of finite rank over K and addition is induced by the tensor product of algebras. It arose out of attempts to classify division algebras over a field and is...

 of K, which is identified with -1 if it is a division algebra and +1 if it is isomorphic to the algebra of 2 by 2 matrices.

Hilbert symbols over the rationals


For a place v of the rational number field and rational numbers a, b we let (a, b)v denote the value of the Hilbert symbol in the corresponding completion
Complete space
In mathematical analysis, a metric space M is called complete if every Cauchy sequence of points in M has a limit that is also in M or, alternatively, if every Cauchy sequence in M converges in M....

 Qv. As usual, if v is the valuation attached to a prime number p then the corresponding completion is the p-adic field
P-adic number
In mathematics, and chiefly number theory, the p-adic number system for any prime number p extends the ordinary arithmetic of the rational numbers in a way different from the extension of the rational number system to the real and complex number systems...

 and if v is the infinite place then the completion is the real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

 field.

Over the reals, (a, b) is +1 if at least one of a or b is positive, and −1 if both are negative.

Over the p-adics with p odd, writing and , where u and v are integers coprime
Coprime
In number theory, a branch of mathematics, two integers a and b are said to be coprime or relatively prime if the only positive integer that evenly divides both of them is 1. This is the same thing as their greatest common divisor being 1...

 to p, we have
, where

and the expression involves two Legendre symbol
Legendre symbol
In number theory, the Legendre symbol is a multiplicative function with values 1, −1, 0 that is a quadratic character modulo a prime number p: its value on a quadratic residue mod p is 1 and on a quadratic non-residue is −1....

s.

Over the 2-adics, again writing and , where u and v are odd numbers, we have
, where .

It is known that if v ranges over all places, (a, b)v is 1 for almost all places. Therefore the following product formula


makes sense. It is equivalent to the law of quadratic reciprocity
Quadratic reciprocity
In number theory, the law of quadratic reciprocity is a theorem about modular arithmetic which gives conditions for the solvability of quadratic equations modulo prime numbers...

.

External links

  • Steinberg symbol at the Encyclopaedia of Mathematics
    Encyclopaedia of Mathematics
    The Encyclopaedia of Mathematics is a large reference work in mathematics. It is available in book form and on CD-ROM....

  • HilbertSymbol at Mathworld
    MathWorld
    MathWorld is an online mathematics reference work, created and largely written by Eric W. Weisstein. It is sponsored by and licensed to Wolfram Research, Inc. and was partially funded by the National Science Foundation's National Science Digital Library grant to the University of Illinois at...